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Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained at large scales by planetary rotation. Rapidly rotating Rayleigh–Bénard convection (RRRBC) provides a paradigm for experiments and direct numerical simulations (DNS) of such flows, but the accessible parameter space remains restricted to moderately fast rotation rates (Ekman numbers$${ {Ek}} \gtrsim 10^{-8}$$), while realistic$${Ek}$$for geo- and astrophysical applications are orders of magnitude smaller. On the other hand, previously derived reduced equations of motion describing the leading-order behaviour in the limit of very rapid rotation ($$ {Ek}\to 0$$) cannot capture finite rotation effects, and the physically most relevant part of parameter space with small but finite$${Ek}$$has remained elusive. Here, we employ the rescaled rapidly rotating incompressible Navier–Stokes equations (RRRiNSE) – a reformulation of the Navier–Stokes–Boussinesq equations informed by the scalings valid for$${Ek}\to 0$$, recently introduced by Julienet al.(2024) – to provide full DNS of RRRBC at unprecedented rotation strengths down to$$ {Ek}=10^{-15}$$and below, revealing the disappearance of cyclone–anticyclone asymmetry at previously unattainable Ekman numbers ($${Ek}\approx 10^{-9}$$). We also identify an overshoot in the heat transport as$${Ek}$$is varied at fixed$$\widetilde { {Ra}} \equiv {Ra}{Ek}^{4/3}$$, where$$Ra$$is the Rayleigh number, associated with dissipation due to ageostrophic motions in the boundary layers. The simulations validate theoretical predictions based on thermal boundary layer theory for RRRBC and show that the solutions of RRRiNSE agree with the reduced equations at very small$${Ek}$$. These results represent a first foray into the vast, largely unexplored parameter space of very rapidly rotating convection rendered accessible by RRRiNSE. 

We study structure formation in two-dimensional turbulence driven by an external force, interpolating between linear instability forcing and random stirring, subject to nonlinear damping. Using extensive direct numerical simulations, we uncover a rich parameter space featuring four distinct branches of stationary solutions: large-scale vortices, hybrid states with embedded shielded vortices (SVs) of either sign, and two states composed of many similar SVs. Of the latter, the first is a dense vortex gas where all SVs have the same sign and diffuse across the domain. The second is a hexagonal vortex crystal forming from this gas when the linear instability is sufficiently weak. These solutions coexist stably over a wide parameter range. The late-time evolution of the system from small-amplitude initial conditions is nearly self-similar, involving three phases: initial inverse cascade, random nucleation of SVs from turbulence and, once a critical number of vortices is reached, a phase of explosive nucleation of SVs, leading to a statistically stationary state. The vortex gas is continued in the forcing parameter, revealing a sharp transition towards the crystal state as the forcing strength decreases. This transition is analysed in terms of the diffusivity of individual vortices using ideas from statistical physics. The crystal can also decay via an inverse cascade resulting from the breakdown of shielding or insufficient nonlinear damping acting on SVs. Our study highlights the importance of the forcing details in two-dimensional turbulence and reveals the presence of non-trivial SV states in this system, specifically the emergence and melting of a vortex crystal. 

This work studies two-dimensional fixed-flux Rayleigh–Bénard convection with periodic boundary conditions in both horizontal and vertical directions and analyses its dynamics using numerical continuation, secondary instability analysis and direct numerical simulation. The fixed-flux constraint leads to time-independent elevator modes with a well-defined amplitude. Secondary instability of these modes leads to tilted elevator modes accompanied by horizontal shear flow. For$$Pr=1$$, where$$Pr$$is the Prandtl number, a subsequent subcritical Hopf bifurcation leads to hysteresis behaviour between this state and a time-dependent direction-reversing state, followed by a global bifurcation leading to modulated travelling waves without flow reversal. Single-mode equations reproduce this moderate Rayleigh number behaviour well. At high Rayleigh numbers, chaotic behaviour dominated by modulated travelling waves appears. These transitions are characteristic of high wavenumber elevator modes since the vertical wavenumber of the secondary instability is linearly proportional to the horizontal wavenumber of the elevator mode. At a low$$Pr$$, relaxation oscillations between the conduction state and the elevator mode appear, followed by quasi-periodic and chaotic behaviour as the Rayleigh number increases. In the high$$Pr$$regime, the large-scale shear weakens, and the flow shows bursting behaviour that can lead to significantly increased heat transport or even intermittent stable stratification. 

Abstract Starting from the fully compressible fluid equations in a plane-parallel atmosphere, we demonstrate that linear internal gravity waves are naturally pseudo-incompressible in the limit that the wave frequencyωis much less than that of surface gravity waves, i.e., $\omega \ll \sqrt{{\mathit{gk}}_h}$, wheregis the gravitational acceleration andk_his the horizontal wavenumber. We accomplish this by performing a formal expansion of the wave functions and the local dispersion relation in terms of a dimensionless frequency $\epsilon =\omega /\sqrt{{\mathit{gk}}_h}$. Further, we show that, in this same low-frequency limit, several forms of the anelastic approximation, including the Lantz–Braginsky–Roberts formulation, poorly reproduce the correct behavior of internal gravity waves. The pseudo-incompressible approximation is achieved by assuming that Eulerian fluctuations of the pressure are small in the continuity equation—whereas, in the anelastic approximation, Eulerian density fluctuations are ignored. In an adiabatic stratification, such as occurs in a convection zone, the two approximations become identical. However, in a stable stratification, the differences between the two approximations are stark and only the pseudo-incompressible approximation remains valid. 

Gyroscopic alignment gives rise to highly spatially anisotropic columnar structures that in combination with complex domain boundaries pose challenges for efficient numerical discretizations and computations. We define gyroscopic polynomials to be three-dimensional polynomials expressed in a coordinate system that conforms to rotational alignment. We remap the original domain with radius-dependent boundaries onto a right cylindrical or annular domain to create the computational domain in this coordinate system. We find the volume element expressed in gyroscopic coordinates leads naturally to a hierarchy of orthonormal bases. We build the bases out of Jacobi polynomials in the vertical and generalized Jacobi polynomials in the radial. Because these coordinates explicitly conform to flow structures found in rapidly rotating systems the bases represent fields with a relatively small number of modes. We develop the operator structure for one-dimensional semi-classical orthogonal polynomials as a building block for differential operators in the full three-dimensional cylindrical and annular domains. The differentiation operators of generalized Jacobi polynomials generate a sparse linear system for discretization of differential operators acting on the gyroscopic bases. This enables efficient simulation of systems with strong gyroscopic alignment. 

A numerical investigation of an asymptotically reduced model for quasigeostrophic Rayleigh-Bénard convection is conducted in which the depth-averaged flows are numerically suppressed by modifying the governing equations. At the largest accessible values of the Rayleigh number Ra, the Reynolds number and Nusselt number show evidence of approaching the diffusion-free scalings of Re ∼ RaE/Pr and Nu ∼ Pr−1/2Ra3/2E2, respectively, where E is the Ekman number and Pr is the Prandtl number. For large Ra, the presence of depth-invariant flows, such as large-scale vortices, yield heat and momentum transport scalings that exceed those of the diffusion-free scaling laws. The Taylor microscale does not vary significantly with increasing Ra, whereas the integral length scale grows weakly. The computed length scales remain O(1) with respect to the linearly unstable critical wave number; we therefore conclude that these scales remain viscously controlled. We do not find a point-wise Coriolis-inertia-Archimedean (CIA) force balance in the turbulent regime; interior dynamics are instead dominated by horizontal advection (inertia), vortex stretching (Coriolis) and the vertical pressure gradient. A secondary, subdominant balance between the Archimedean buoyancy force and the viscous force occurs in the interior and the ratio of the root mean square (rms) of these two forces is found to approach unity with increasing Ra. This secondary balance is attributed to the turbulent fluid interior acting as the dominant control on the heat transport. These findings indicate that a pointwise CIA balance does not occur in the high Rayleigh number regime of quasigeostrophic convection in the plane layer geometry. Instead, simulations are characterized by what may be termed a nonlocal CIA balance in which the buoyancy force is dominant within the thermal boundary layers and is spatially separated from the interior Coriolis and inertial forces. 

Bifurcation analysis of confined salt-finger convection using single-mode equations obtained from a severely truncated Fourier expansion in the horizontal is performed. Strongly nonlinear staircase-like solutions having, respectively, one (S1), two (S2) and three (S3) regions of mixed salinity in the vertical direction are computed using numerical continuation, and their stability properties are determined. Near onset, the one-layer S1 solution is stable and corresponds to maximum salinity transport among the three solutions. The S2 and S3 solutions are unstable but exert an influence on the statistics observed in direct numerical simulations (DNS) in larger two-dimensional (2-D) domains. Secondary bifurcations of S1 lead either to tilted-finger (TF1) or to travelling wave (TW1) solutions, both accompanied by the spontaneous generation of large-scale shear, a process favoured for lower density ratios and Prandtl numbers ( $Pr$ ). These states at low $Pr$ are associated, respectively, with two-layer and three-layer staircase-like salinity profiles in the mean. States breaking reflection symmetry in the midplane are also computed. In two dimensions and for low $Pr$ , the DNS results favour direction-reversing tilted fingers resembling the pulsating wave state observed in other systems. Two-layer and three-layer mean salinity profiles corresponding to reversing tilted fingers and TW1 are observed in 2-D DNS averaged over time. The single-mode solutions close to the high wavenumber onset are in an excellent agreement with 2-D DNS in small horizontal domains and compare well with 3-D DNS. 

Instabilities of fluid flows often generate turbulence. Using extensive direct numerical simulations, we study two-dimensional turbulence driven by a wavenumber-localised instability superposed on stochastic forcing, in contrast to previous studies of state-independent forcing. As the contribution of the instability forcing, measured by a parameter $$\gamma$$ , increases, the system undergoes two transitions. For $$\gamma$$ below a first threshold, a regular large-scale vortex condensate forms. Above this threshold, shielded vortices (SVs) emerge within the condensate. At a second, larger value of $$\gamma$$ , the condensate breaks down, and a gas of weakly interacting vortices with broken symmetry spontaneously emerges, characterised by preponderance of vortices of one sign only and suppressed inverse energy cascade. The latter transition is shown to depend on the damping mechanism. The number density of SVs in the broken symmetry state slowly increases via a random nucleation process. Bistability is observed between the condensate and mixed SV-condensate states. Our findings provide new evidence for a strong dependence of two-dimensional turbulence phenomenology on the forcing. 

The connection between the heat transfer and characteristic flow velocities of planetary core-style convection remains poorly understood. To address this, we present novel laboratory models of rotating Rayleigh–Bénard convection in which heat and momentum transfer are simultaneously measured. Using water (Prandtl number, Pr≃6) and cylindrical containers of diameter-to-height aspect ratios of Γ≃3,1.5,0.75, the non-dimensional rotation period (Ekman number, E) is varied between 10−7≲E≲3×10−5 and the non-dimensional convective forcing (Rayleigh number, Ra) ranges from 107≲Ra≲1012. Our heat transfer data agree with those of previous studies and are largely controlled by boundary layer dynamics. We utilize laser Doppler velocimetry (LDV) to obtain experimental point measurements of bulk axial velocities, resulting in estimates of the non-dimensional momentum transfer (Reynolds number, Re) with values between 4×102≲Re≲5×104. Behavioral transitions in the velocity data do not exist where transitions in heat transfer behaviors occur, indicating that bulk dynamics are not controlled by the boundary layers of the system. Instead, the LDV data agree well with the diffusion-free Coriolis–Inertia–Archimedian (CIA) scaling over the range of Ra explored. Furthermore, the CIA scaling approximately co-scales with the Viscous–Archimedian–Coriolis (VAC) scaling over the parameter space studied. We explain this observation by demonstrating that the VAC and CIA relations will co-scale when the local Reynolds number in the fluid bulk is of order unity. We conclude that in our experiments and similar laboratory and numerical investigations with E≳10−7, Ra≲1012, Pr≃7, heat transfer is controlled by boundary layer physics while quasi-geostrophically turbulent dynamics relevant to core flows robustly exist in the fluid bulk.